Shadow Lugia, also known as XD001, is an experiment by Cipher to make an UnPurifyable totally evil Pokémon to cater to their every whim. Cipher first used it to kidnap the Crusier Libra so that they could harvest the Pokémon the travellers left when they evacuated the Libra

Your encounter with Shadow Lugia is when you battle with DeathGold. It is at Level 50. Don't worry if you accidentally knock it out. You can always come back afterwards and Snag him again

Once you have snagged Shadow Lugia you are free to use him in battle. However Battling, using Cologne or placing it in a Breeding Centre will not work in the purification process. For that you need to use the Purification Chamber. But in the mean time here are Shadow Lugia's Attacks when he's still Shadowed

The only way to purify Shadow Lugia is to put it in the Purification Chamber. You need all 9 rooms of the chamber to have completely maxed out tempo...but it doesn't matter if Pokémon are being purified within them. All you need to do is put Lugia in one of the rooms in the middle and he'll be instantly purified

Now he's purified, you will be able to use the standard Lugia in battle. He looks fully normal and you'll be able to trade him to your Game Boy games. You'll be able top level him up like all Pokémon. It's completely yours